3.179 \(\int x (a+b x)^n (c+d x^3)^2 \, dx\)

Optimal. Leaf size=248 \[ -\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^8 (n+1)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{n+4}}{b^8 (n+4)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac {21 a^2 d^2 (a+b x)^{n+6}}{b^8 (n+6)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac {7 a d^2 (a+b x)^{n+7}}{b^8 (n+7)}+\frac {d^2 (a+b x)^{n+8}}{b^8 (n+8)} \]

[Out]

-a*(-a^3*d+b^3*c)^2*(b*x+a)^(1+n)/b^8/(1+n)+(-7*a^3*d+b^3*c)*(-a^3*d+b^3*c)*(b*x+a)^(2+n)/b^8/(2+n)+3*a^2*d*(-
7*a^3*d+4*b^3*c)*(b*x+a)^(3+n)/b^8/(3+n)-a*d*(-35*a^3*d+8*b^3*c)*(b*x+a)^(4+n)/b^8/(4+n)+d*(-35*a^3*d+2*b^3*c)
*(b*x+a)^(5+n)/b^8/(5+n)+21*a^2*d^2*(b*x+a)^(6+n)/b^8/(6+n)-7*a*d^2*(b*x+a)^(7+n)/b^8/(7+n)+d^2*(b*x+a)^(8+n)/
b^8/(8+n)

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Rubi [A]  time = 0.15, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1620} \[ -\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^8 (n+1)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^8 (n+2)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{n+4}}{b^8 (n+4)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac {21 a^2 d^2 (a+b x)^{n+6}}{b^8 (n+6)}-\frac {7 a d^2 (a+b x)^{n+7}}{b^8 (n+7)}+\frac {d^2 (a+b x)^{n+8}}{b^8 (n+8)} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^n*(c + d*x^3)^2,x]

[Out]

-((a*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^3*c - 7*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(2 + n
))/(b^8*(2 + n)) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) - (a*d*(8*b^3*c - 35*a^3*d)*(
a + b*x)^(4 + n))/(b^8*(4 + n)) + (d*(2*b^3*c - 35*a^3*d)*(a + b*x)^(5 + n))/(b^8*(5 + n)) + (21*a^2*d^2*(a +
b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^2*(a + b*x)^(7 + n))/(b^8*(7 + n)) + (d^2*(a + b*x)^(8 + n))/(b^8*(8 + n)
)

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {align*} \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx &=\int \left (-\frac {a \left (-b^3 c+a^3 d\right )^2 (a+b x)^n}{b^7}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^7}-\frac {3 a^2 d \left (-4 b^3 c+7 a^3 d\right ) (a+b x)^{2+n}}{b^7}+\frac {a d \left (-8 b^3 c+35 a^3 d\right ) (a+b x)^{3+n}}{b^7}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{4+n}}{b^7}+\frac {21 a^2 d^2 (a+b x)^{5+n}}{b^7}-\frac {7 a d^2 (a+b x)^{6+n}}{b^7}+\frac {d^2 (a+b x)^{7+n}}{b^7}\right ) \, dx\\ &=-\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^8 (1+n)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^8 (2+n)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{3+n}}{b^8 (3+n)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{4+n}}{b^8 (4+n)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{5+n}}{b^8 (5+n)}+\frac {21 a^2 d^2 (a+b x)^{6+n}}{b^8 (6+n)}-\frac {7 a d^2 (a+b x)^{7+n}}{b^8 (7+n)}+\frac {d^2 (a+b x)^{8+n}}{b^8 (8+n)}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 211, normalized size = 0.85 \[ \frac {(a+b x)^{n+1} \left (\frac {d (a+b x)^4 \left (2 b^3 c-35 a^3 d\right )}{n+5}+\frac {a d (a+b x)^3 \left (35 a^3 d-8 b^3 c\right )}{n+4}+\frac {(a+b x) \left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right )}{n+2}-\frac {a \left (b^3 c-a^3 d\right )^2}{n+1}+\frac {21 a^2 d^2 (a+b x)^5}{n+6}+\frac {3 a^2 d (a+b x)^2 \left (4 b^3 c-7 a^3 d\right )}{n+3}+\frac {d^2 (a+b x)^7}{n+8}-\frac {7 a d^2 (a+b x)^6}{n+7}\right )}{b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^n*(c + d*x^3)^2,x]

[Out]

((a + b*x)^(1 + n)*(-((a*(b^3*c - a^3*d)^2)/(1 + n)) + ((b^3*c - 7*a^3*d)*(b^3*c - a^3*d)*(a + b*x))/(2 + n) +
(3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x)^2)/(3 + n) + (a*d*(-8*b^3*c + 35*a^3*d)*(a + b*x)^3)/(4 + n) + (d*(2*b
^3*c - 35*a^3*d)*(a + b*x)^4)/(5 + n) + (21*a^2*d^2*(a + b*x)^5)/(6 + n) - (7*a*d^2*(a + b*x)^6)/(7 + n) + (d^
2*(a + b*x)^7)/(8 + n)))/b^8

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fricas [B]  time = 0.53, size = 1216, normalized size = 4.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="fricas")

[Out]

-(a^2*b^6*c^2*n^6 + 33*a^2*b^6*c^2*n^5 + 445*a^2*b^6*c^2*n^4 + 20160*a^2*b^6*c^2 - 16128*a^5*b^3*c*d + 5040*a^
8*d^2 - (b^8*d^2*n^7 + 28*b^8*d^2*n^6 + 322*b^8*d^2*n^5 + 1960*b^8*d^2*n^4 + 6769*b^8*d^2*n^3 + 13132*b^8*d^2*
n^2 + 13068*b^8*d^2*n + 5040*b^8*d^2)*x^8 - (a*b^7*d^2*n^7 + 21*a*b^7*d^2*n^6 + 175*a*b^7*d^2*n^5 + 735*a*b^7*
d^2*n^4 + 1624*a*b^7*d^2*n^3 + 1764*a*b^7*d^2*n^2 + 720*a*b^7*d^2*n)*x^7 + 7*(a^2*b^6*d^2*n^6 + 15*a^2*b^6*d^2
*n^5 + 85*a^2*b^6*d^2*n^4 + 225*a^2*b^6*d^2*n^3 + 274*a^2*b^6*d^2*n^2 + 120*a^2*b^6*d^2*n)*x^6 - 2*(b^8*c*d*n^
7 + 31*b^8*c*d*n^6 + 8064*b^8*c*d + (391*b^8*c*d + 21*a^3*b^5*d^2)*n^5 + (2581*b^8*c*d + 210*a^3*b^5*d^2)*n^4
+ (9544*b^8*c*d + 735*a^3*b^5*d^2)*n^3 + 2*(9782*b^8*c*d + 525*a^3*b^5*d^2)*n^2 + 72*(282*b^8*c*d + 7*a^3*b^5*
d^2)*n)*x^5 - 2*(a*b^7*c*d*n^7 + 27*a*b^7*c*d*n^6 + 283*a*b^7*c*d*n^5 + 21*(69*a*b^7*c*d - 5*a^4*b^4*d^2)*n^4
+ 2*(1874*a*b^7*c*d - 315*a^4*b^4*d^2)*n^3 + 3*(1524*a*b^7*c*d - 385*a^4*b^4*d^2)*n^2 + 126*(16*a*b^7*c*d - 5*
a^4*b^4*d^2)*n)*x^4 + 3*(1045*a^2*b^6*c^2 - 16*a^5*b^3*c*d)*n^3 + 8*(a^2*b^6*c*d*n^6 + 24*a^2*b^6*c*d*n^5 + 21
1*a^2*b^6*c*d*n^4 + 3*(272*a^2*b^6*c*d - 35*a^5*b^3*d^2)*n^3 + 5*(260*a^2*b^6*c*d - 63*a^5*b^3*d^2)*n^2 + 42*(
16*a^2*b^6*c*d - 5*a^5*b^3*d^2)*n)*x^3 + 2*(6077*a^2*b^6*c^2 - 504*a^5*b^3*c*d)*n^2 - (b^8*c^2*n^7 + 34*b^8*c^
2*n^6 + 20160*b^8*c^2 + 2*(239*b^8*c^2 + 12*a^3*b^5*c*d)*n^5 + 4*(895*b^8*c^2 + 132*a^3*b^5*c*d)*n^4 + (15289*
b^8*c^2 + 4008*a^3*b^5*c*d)*n^3 + 2*(18353*b^8*c^2 + 5784*a^3*b^5*c*d - 1260*a^6*b^2*d^2)*n^2 + 72*(621*b^8*c^
2 + 112*a^3*b^5*c*d - 35*a^6*b^2*d^2)*n)*x^2 + 24*(1023*a^2*b^6*c^2 - 292*a^5*b^3*c*d)*n - (a*b^7*c^2*n^7 + 33
*a*b^7*c^2*n^6 + 445*a*b^7*c^2*n^5 + 3*(1045*a*b^7*c^2 - 16*a^4*b^4*c*d)*n^4 + 2*(6077*a*b^7*c^2 - 504*a^4*b^4
*c*d)*n^3 + 24*(1023*a*b^7*c^2 - 292*a^4*b^4*c*d)*n^2 + 1008*(20*a*b^7*c^2 - 16*a^4*b^4*c*d + 5*a^7*b*d^2)*n)*
x)*(b*x + a)^n/(b^8*n^8 + 36*b^8*n^7 + 546*b^8*n^6 + 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8
*n^2 + 109584*b^8*n + 40320*b^8)

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giac [B]  time = 0.61, size = 2034, normalized size = 8.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="giac")

[Out]

((b*x + a)^n*b^8*d^2*n^7*x^8 + (b*x + a)^n*a*b^7*d^2*n^7*x^7 + 28*(b*x + a)^n*b^8*d^2*n^6*x^8 + 21*(b*x + a)^n
*a*b^7*d^2*n^6*x^7 + 322*(b*x + a)^n*b^8*d^2*n^5*x^8 + 2*(b*x + a)^n*b^8*c*d*n^7*x^5 - 7*(b*x + a)^n*a^2*b^6*d
^2*n^6*x^6 + 175*(b*x + a)^n*a*b^7*d^2*n^5*x^7 + 1960*(b*x + a)^n*b^8*d^2*n^4*x^8 + 2*(b*x + a)^n*a*b^7*c*d*n^
7*x^4 + 62*(b*x + a)^n*b^8*c*d*n^6*x^5 - 105*(b*x + a)^n*a^2*b^6*d^2*n^5*x^6 + 735*(b*x + a)^n*a*b^7*d^2*n^4*x
^7 + 6769*(b*x + a)^n*b^8*d^2*n^3*x^8 + 54*(b*x + a)^n*a*b^7*c*d*n^6*x^4 + 782*(b*x + a)^n*b^8*c*d*n^5*x^5 + 4
2*(b*x + a)^n*a^3*b^5*d^2*n^5*x^5 - 595*(b*x + a)^n*a^2*b^6*d^2*n^4*x^6 + 1624*(b*x + a)^n*a*b^7*d^2*n^3*x^7 +
13132*(b*x + a)^n*b^8*d^2*n^2*x^8 + (b*x + a)^n*b^8*c^2*n^7*x^2 - 8*(b*x + a)^n*a^2*b^6*c*d*n^6*x^3 + 566*(b*
x + a)^n*a*b^7*c*d*n^5*x^4 + 5162*(b*x + a)^n*b^8*c*d*n^4*x^5 + 420*(b*x + a)^n*a^3*b^5*d^2*n^4*x^5 - 1575*(b*
x + a)^n*a^2*b^6*d^2*n^3*x^6 + 1764*(b*x + a)^n*a*b^7*d^2*n^2*x^7 + 13068*(b*x + a)^n*b^8*d^2*n*x^8 + (b*x + a
)^n*a*b^7*c^2*n^7*x + 34*(b*x + a)^n*b^8*c^2*n^6*x^2 - 192*(b*x + a)^n*a^2*b^6*c*d*n^5*x^3 + 2898*(b*x + a)^n*
a*b^7*c*d*n^4*x^4 - 210*(b*x + a)^n*a^4*b^4*d^2*n^4*x^4 + 19088*(b*x + a)^n*b^8*c*d*n^3*x^5 + 1470*(b*x + a)^n
*a^3*b^5*d^2*n^3*x^5 - 1918*(b*x + a)^n*a^2*b^6*d^2*n^2*x^6 + 720*(b*x + a)^n*a*b^7*d^2*n*x^7 + 5040*(b*x + a)
^n*b^8*d^2*x^8 + 33*(b*x + a)^n*a*b^7*c^2*n^6*x + 478*(b*x + a)^n*b^8*c^2*n^5*x^2 + 24*(b*x + a)^n*a^3*b^5*c*d
*n^5*x^2 - 1688*(b*x + a)^n*a^2*b^6*c*d*n^4*x^3 + 7496*(b*x + a)^n*a*b^7*c*d*n^3*x^4 - 1260*(b*x + a)^n*a^4*b^
4*d^2*n^3*x^4 + 39128*(b*x + a)^n*b^8*c*d*n^2*x^5 + 2100*(b*x + a)^n*a^3*b^5*d^2*n^2*x^5 - 840*(b*x + a)^n*a^2
*b^6*d^2*n*x^6 - (b*x + a)^n*a^2*b^6*c^2*n^6 + 445*(b*x + a)^n*a*b^7*c^2*n^5*x + 3580*(b*x + a)^n*b^8*c^2*n^4*
x^2 + 528*(b*x + a)^n*a^3*b^5*c*d*n^4*x^2 - 6528*(b*x + a)^n*a^2*b^6*c*d*n^3*x^3 + 840*(b*x + a)^n*a^5*b^3*d^2
*n^3*x^3 + 9144*(b*x + a)^n*a*b^7*c*d*n^2*x^4 - 2310*(b*x + a)^n*a^4*b^4*d^2*n^2*x^4 + 40608*(b*x + a)^n*b^8*c
*d*n*x^5 + 1008*(b*x + a)^n*a^3*b^5*d^2*n*x^5 - 33*(b*x + a)^n*a^2*b^6*c^2*n^5 + 3135*(b*x + a)^n*a*b^7*c^2*n^
4*x - 48*(b*x + a)^n*a^4*b^4*c*d*n^4*x + 15289*(b*x + a)^n*b^8*c^2*n^3*x^2 + 4008*(b*x + a)^n*a^3*b^5*c*d*n^3*
x^2 - 10400*(b*x + a)^n*a^2*b^6*c*d*n^2*x^3 + 2520*(b*x + a)^n*a^5*b^3*d^2*n^2*x^3 + 4032*(b*x + a)^n*a*b^7*c*
d*n*x^4 - 1260*(b*x + a)^n*a^4*b^4*d^2*n*x^4 + 16128*(b*x + a)^n*b^8*c*d*x^5 - 445*(b*x + a)^n*a^2*b^6*c^2*n^4
+ 12154*(b*x + a)^n*a*b^7*c^2*n^3*x - 1008*(b*x + a)^n*a^4*b^4*c*d*n^3*x + 36706*(b*x + a)^n*b^8*c^2*n^2*x^2
+ 11568*(b*x + a)^n*a^3*b^5*c*d*n^2*x^2 - 2520*(b*x + a)^n*a^6*b^2*d^2*n^2*x^2 - 5376*(b*x + a)^n*a^2*b^6*c*d*
n*x^3 + 1680*(b*x + a)^n*a^5*b^3*d^2*n*x^3 - 3135*(b*x + a)^n*a^2*b^6*c^2*n^3 + 48*(b*x + a)^n*a^5*b^3*c*d*n^3
+ 24552*(b*x + a)^n*a*b^7*c^2*n^2*x - 7008*(b*x + a)^n*a^4*b^4*c*d*n^2*x + 44712*(b*x + a)^n*b^8*c^2*n*x^2 +
8064*(b*x + a)^n*a^3*b^5*c*d*n*x^2 - 2520*(b*x + a)^n*a^6*b^2*d^2*n*x^2 - 12154*(b*x + a)^n*a^2*b^6*c^2*n^2 +
1008*(b*x + a)^n*a^5*b^3*c*d*n^2 + 20160*(b*x + a)^n*a*b^7*c^2*n*x - 16128*(b*x + a)^n*a^4*b^4*c*d*n*x + 5040*
(b*x + a)^n*a^7*b*d^2*n*x + 20160*(b*x + a)^n*b^8*c^2*x^2 - 24552*(b*x + a)^n*a^2*b^6*c^2*n + 7008*(b*x + a)^n
*a^5*b^3*c*d*n - 20160*(b*x + a)^n*a^2*b^6*c^2 + 16128*(b*x + a)^n*a^5*b^3*c*d - 5040*(b*x + a)^n*a^8*d^2)/(b^
8*n^8 + 36*b^8*n^7 + 546*b^8*n^6 + 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8*n^2 + 109584*b^8*
n + 40320*b^8)

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maple [B]  time = 0.02, size = 1142, normalized size = 4.60 \[ -\frac {\left (-b^{7} d^{2} n^{7} x^{7}-28 b^{7} d^{2} n^{6} x^{7}+7 a \,b^{6} d^{2} n^{6} x^{6}-322 b^{7} d^{2} n^{5} x^{7}+147 a \,b^{6} d^{2} n^{5} x^{6}-2 b^{7} c d \,n^{7} x^{4}-1960 b^{7} d^{2} n^{4} x^{7}-42 a^{2} b^{5} d^{2} n^{5} x^{5}+1225 a \,b^{6} d^{2} n^{4} x^{6}-62 b^{7} c d \,n^{6} x^{4}-6769 b^{7} d^{2} n^{3} x^{7}-630 a^{2} b^{5} d^{2} n^{4} x^{5}+8 a \,b^{6} c d \,n^{6} x^{3}+5145 a \,b^{6} d^{2} n^{3} x^{6}-782 b^{7} c d \,n^{5} x^{4}-13132 b^{7} d^{2} n^{2} x^{7}+210 a^{3} b^{4} d^{2} n^{4} x^{4}-3570 a^{2} b^{5} d^{2} n^{3} x^{5}+216 a \,b^{6} c d \,n^{5} x^{3}+11368 a \,b^{6} d^{2} n^{2} x^{6}-b^{7} c^{2} n^{7} x -5162 b^{7} c d \,n^{4} x^{4}-13068 b^{7} d^{2} n \,x^{7}+2100 a^{3} b^{4} d^{2} n^{3} x^{4}-24 a^{2} b^{5} c d \,n^{5} x^{2}-9450 a^{2} b^{5} d^{2} n^{2} x^{5}+2264 a \,b^{6} c d \,n^{4} x^{3}+12348 a \,b^{6} d^{2} n \,x^{6}-34 b^{7} c^{2} n^{6} x -19088 b^{7} c d \,n^{3} x^{4}-5040 d^{2} x^{7} b^{7}-840 a^{4} b^{3} d^{2} n^{3} x^{3}+7350 a^{3} b^{4} d^{2} n^{2} x^{4}-576 a^{2} b^{5} c d \,n^{4} x^{2}-11508 a^{2} b^{5} d^{2} n \,x^{5}+a \,b^{6} c^{2} n^{6}+11592 a \,b^{6} c d \,n^{3} x^{3}+5040 a \,d^{2} x^{6} b^{6}-478 b^{7} c^{2} n^{5} x -39128 b^{7} c d \,n^{2} x^{4}-5040 a^{4} b^{3} d^{2} n^{2} x^{3}+48 a^{3} b^{4} c d \,n^{4} x +10500 a^{3} b^{4} d^{2} n \,x^{4}-5064 a^{2} b^{5} c d \,n^{3} x^{2}-5040 a^{2} d^{2} x^{5} b^{5}+33 a \,b^{6} c^{2} n^{5}+29984 a \,b^{6} c d \,n^{2} x^{3}-3580 b^{7} c^{2} n^{4} x -40608 b^{7} c d n \,x^{4}+2520 a^{5} b^{2} d^{2} n^{2} x^{2}-9240 a^{4} b^{3} d^{2} n \,x^{3}+1056 a^{3} b^{4} c d \,n^{3} x +5040 a^{3} b^{4} d^{2} x^{4}-19584 a^{2} b^{5} c d \,n^{2} x^{2}+445 a \,b^{6} c^{2} n^{4}+36576 a \,b^{6} c d n \,x^{3}-15289 b^{7} c^{2} n^{3} x -16128 b^{7} c d \,x^{4}+7560 a^{5} b^{2} d^{2} n \,x^{2}-48 a^{4} b^{3} c d \,n^{3}-5040 a^{4} b^{3} d^{2} x^{3}+8016 a^{3} b^{4} c d \,n^{2} x -31200 a^{2} b^{5} c d n \,x^{2}+3135 a \,b^{6} c^{2} n^{3}+16128 a \,b^{6} c d \,x^{3}-36706 b^{7} c^{2} n^{2} x -5040 a^{6} b \,d^{2} n x +5040 a^{5} b^{2} d^{2} x^{2}-1008 a^{4} b^{3} c d \,n^{2}+23136 a^{3} b^{4} c d n x -16128 a^{2} b^{5} c d \,x^{2}+12154 a \,b^{6} c^{2} n^{2}-44712 b^{7} c^{2} n x -5040 a^{6} b \,d^{2} x -7008 a^{4} b^{3} c d n +16128 a^{3} b^{4} c d x +24552 a \,b^{6} c^{2} n -20160 b^{7} c^{2} x +5040 a^{7} d^{2}-16128 a^{4} b^{3} c d +20160 a \,b^{6} c^{2}\right ) \left (b x +a \right )^{n +1}}{\left (n^{8}+36 n^{7}+546 n^{6}+4536 n^{5}+22449 n^{4}+67284 n^{3}+118124 n^{2}+109584 n +40320\right ) b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^n*(d*x^3+c)^2,x)

[Out]

-(b*x+a)^(n+1)*(-b^7*d^2*n^7*x^7-28*b^7*d^2*n^6*x^7+7*a*b^6*d^2*n^6*x^6-322*b^7*d^2*n^5*x^7+147*a*b^6*d^2*n^5*
x^6-2*b^7*c*d*n^7*x^4-1960*b^7*d^2*n^4*x^7-42*a^2*b^5*d^2*n^5*x^5+1225*a*b^6*d^2*n^4*x^6-62*b^7*c*d*n^6*x^4-67
69*b^7*d^2*n^3*x^7-630*a^2*b^5*d^2*n^4*x^5+8*a*b^6*c*d*n^6*x^3+5145*a*b^6*d^2*n^3*x^6-782*b^7*c*d*n^5*x^4-1313
2*b^7*d^2*n^2*x^7+210*a^3*b^4*d^2*n^4*x^4-3570*a^2*b^5*d^2*n^3*x^5+216*a*b^6*c*d*n^5*x^3+11368*a*b^6*d^2*n^2*x
^6-b^7*c^2*n^7*x-5162*b^7*c*d*n^4*x^4-13068*b^7*d^2*n*x^7+2100*a^3*b^4*d^2*n^3*x^4-24*a^2*b^5*c*d*n^5*x^2-9450
*a^2*b^5*d^2*n^2*x^5+2264*a*b^6*c*d*n^4*x^3+12348*a*b^6*d^2*n*x^6-34*b^7*c^2*n^6*x-19088*b^7*c*d*n^3*x^4-5040*
b^7*d^2*x^7-840*a^4*b^3*d^2*n^3*x^3+7350*a^3*b^4*d^2*n^2*x^4-576*a^2*b^5*c*d*n^4*x^2-11508*a^2*b^5*d^2*n*x^5+a
*b^6*c^2*n^6+11592*a*b^6*c*d*n^3*x^3+5040*a*b^6*d^2*x^6-478*b^7*c^2*n^5*x-39128*b^7*c*d*n^2*x^4-5040*a^4*b^3*d
^2*n^2*x^3+48*a^3*b^4*c*d*n^4*x+10500*a^3*b^4*d^2*n*x^4-5064*a^2*b^5*c*d*n^3*x^2-5040*a^2*b^5*d^2*x^5+33*a*b^6
*c^2*n^5+29984*a*b^6*c*d*n^2*x^3-3580*b^7*c^2*n^4*x-40608*b^7*c*d*n*x^4+2520*a^5*b^2*d^2*n^2*x^2-9240*a^4*b^3*
d^2*n*x^3+1056*a^3*b^4*c*d*n^3*x+5040*a^3*b^4*d^2*x^4-19584*a^2*b^5*c*d*n^2*x^2+445*a*b^6*c^2*n^4+36576*a*b^6*
c*d*n*x^3-15289*b^7*c^2*n^3*x-16128*b^7*c*d*x^4+7560*a^5*b^2*d^2*n*x^2-48*a^4*b^3*c*d*n^3-5040*a^4*b^3*d^2*x^3
+8016*a^3*b^4*c*d*n^2*x-31200*a^2*b^5*c*d*n*x^2+3135*a*b^6*c^2*n^3+16128*a*b^6*c*d*x^3-36706*b^7*c^2*n^2*x-504
0*a^6*b*d^2*n*x+5040*a^5*b^2*d^2*x^2-1008*a^4*b^3*c*d*n^2+23136*a^3*b^4*c*d*n*x-16128*a^2*b^5*c*d*x^2+12154*a*
b^6*c^2*n^2-44712*b^7*c^2*n*x-5040*a^6*b*d^2*x-7008*a^4*b^3*c*d*n+16128*a^3*b^4*c*d*x+24552*a*b^6*c^2*n-20160*
b^7*c^2*x+5040*a^7*d^2-16128*a^4*b^3*c*d+20160*a*b^6*c^2)/b^8/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3
+118124*n^2+109584*n+40320)

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maxima [A]  time = 0.98, size = 474, normalized size = 1.91 \[ \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {{\left ({\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{8} x^{8} + {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a b^{7} x^{7} - 7 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{2} b^{6} x^{6} + 42 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{3} b^{5} x^{5} - 210 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{4} b^{4} x^{4} + 840 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{5} b^{3} x^{3} - 2520 \, {\left (n^{2} + n\right )} a^{6} b^{2} x^{2} + 5040 \, a^{7} b n x - 5040 \, a^{8}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="maxima")

[Out]

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((n^4 + 10*n^3 + 35*n^2 + 50*n + 2
4)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2
*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*c*d/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5) + ((n^7 +
28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n
^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*b^7*x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^2*b^6*x^
6 + 42*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^3*b^5*x^5 - 210*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^4*b^4*x^4 + 84
0*(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 - 2520*(n^2 + n)*a^6*b^2*x^2 + 5040*a^7*b*n*x - 5040*a^8)*(b*x + a)^n*d^2/((
n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 40320)*b^8)

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mupad [B]  time = 3.39, size = 1136, normalized size = 4.58 \[ \frac {d^2\,x^8\,{\left (a+b\,x\right )}^n\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}{n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320}-\frac {a^2\,{\left (a+b\,x\right )}^n\,\left (5040\,a^6\,d^2-48\,a^3\,b^3\,c\,d\,n^3-1008\,a^3\,b^3\,c\,d\,n^2-7008\,a^3\,b^3\,c\,d\,n-16128\,a^3\,b^3\,c\,d+b^6\,c^2\,n^6+33\,b^6\,c^2\,n^5+445\,b^6\,c^2\,n^4+3135\,b^6\,c^2\,n^3+12154\,b^6\,c^2\,n^2+24552\,b^6\,c^2\,n+20160\,b^6\,c^2\right )}{b^8\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-2520\,a^6\,d^2\,n+24\,a^3\,b^3\,c\,d\,n^4+504\,a^3\,b^3\,c\,d\,n^3+3504\,a^3\,b^3\,c\,d\,n^2+8064\,a^3\,b^3\,c\,d\,n+b^6\,c^2\,n^6+33\,b^6\,c^2\,n^5+445\,b^6\,c^2\,n^4+3135\,b^6\,c^2\,n^3+12154\,b^6\,c^2\,n^2+24552\,b^6\,c^2\,n+20160\,b^6\,c^2\right )}{b^6\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (5040\,a^6\,d^2-48\,a^3\,b^3\,c\,d\,n^3-1008\,a^3\,b^3\,c\,d\,n^2-7008\,a^3\,b^3\,c\,d\,n-16128\,a^3\,b^3\,c\,d+b^6\,c^2\,n^6+33\,b^6\,c^2\,n^5+445\,b^6\,c^2\,n^4+3135\,b^6\,c^2\,n^3+12154\,b^6\,c^2\,n^2+24552\,b^6\,c^2\,n+20160\,b^6\,c^2\right )}{b^7\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {2\,d\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )\,\left (21\,d\,a^3\,n+c\,b^3\,n^3+21\,c\,b^3\,n^2+146\,c\,b^3\,n+336\,c\,b^3\right )}{b^3\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {a\,d^2\,n\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{b\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {7\,a^2\,d^2\,n\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b^2\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}+\frac {2\,a\,d\,n\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (-105\,d\,a^3+c\,b^3\,n^3+21\,c\,b^3\,n^2+146\,c\,b^3\,n+336\,c\,b^3\right )}{b^4\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )}-\frac {8\,a^2\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-105\,d\,a^3+c\,b^3\,n^3+21\,c\,b^3\,n^2+146\,c\,b^3\,n+336\,c\,b^3\right )}{b^5\,\left (n^8+36\,n^7+546\,n^6+4536\,n^5+22449\,n^4+67284\,n^3+118124\,n^2+109584\,n+40320\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c + d*x^3)^2*(a + b*x)^n,x)

[Out]

(d^2*x^8*(a + b*x)^n*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))/(109584*n +
118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320) - (a^2*(a + b*x)^n*(5040*a^6*d
^2 + 20160*b^6*c^2 + 24552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3 + 445*b^6*c^2*n^4 + 33*b^6*c^2*n^5
+ b^6*c^2*n^6 - 16128*a^3*b^3*c*d - 7008*a^3*b^3*c*d*n - 1008*a^3*b^3*c*d*n^2 - 48*a^3*b^3*c*d*n^3))/(b^8*(10
9584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (x^2*(n + 1)*(a +
b*x)^n*(20160*b^6*c^2 - 2520*a^6*d^2*n + 24552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3 + 445*b^6*c^2*
n^4 + 33*b^6*c^2*n^5 + b^6*c^2*n^6 + 8064*a^3*b^3*c*d*n + 3504*a^3*b^3*c*d*n^2 + 504*a^3*b^3*c*d*n^3 + 24*a^3*
b^3*c*d*n^4))/(b^6*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)
) + (a*n*x*(a + b*x)^n*(5040*a^6*d^2 + 20160*b^6*c^2 + 24552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3
+ 445*b^6*c^2*n^4 + 33*b^6*c^2*n^5 + b^6*c^2*n^6 - 16128*a^3*b^3*c*d - 7008*a^3*b^3*c*d*n - 1008*a^3*b^3*c*d*n
^2 - 48*a^3*b^3*c*d*n^3))/(b^7*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 +
n^8 + 40320)) + (2*d*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)*(336*b^3*c + 21*b^3*c*n^2 + b^3*c*n^3
+ 21*a^3*d*n + 146*b^3*c*n))/(b^3*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^
7 + n^8 + 40320)) + (a*d^2*n*x^7*(a + b*x)^n*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(b*
(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (7*a^2*d^2*n*x^
6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(b^2*(109584*n + 118124*n^2 + 67284*n^3 + 22449
*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (2*a*d*n*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(336*b^
3*c - 105*a^3*d + 21*b^3*c*n^2 + b^3*c*n^3 + 146*b^3*c*n))/(b^4*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4
+ 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (8*a^2*d*n*x^3*(a + b*x)^n*(3*n + n^2 + 2)*(336*b^3*c - 105*a
^3*d + 21*b^3*c*n^2 + b^3*c*n^3 + 146*b^3*c*n))/(b^5*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5
+ 546*n^6 + 36*n^7 + n^8 + 40320))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**n*(d*x**3+c)**2,x)

[Out]

Timed out

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